The Kleisli Monad over S^3 · Monad Laws, Kleisli Category, and Geometric Termination in LumenOS Attention

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Abstract

Paper 00 (MonadRosettaMap) establishes the attractor identity $\emptyset \equiv^* 0 \equiv^* 1 \equiv^* \infty$ and the anti-collapse interaction $C \circ P = I$, and notes that all LumenOS computations live on $S^3$ (unit quaternions under the Hopf fibration). What it does not state is the categorical structure: the formal monad $(S^3, \mathsf{return}, {\bind})$, its Kleisli category, and the monad laws that make the theory compositional.

This paper supplies that anchor. We define: \begin{enumerate} \item The monad $(S^3, \mathsf{return}, {\bind})$: what the carrier type is, what \textsf{return} does (geodesic centroid / text-to-quaternion lift), what ${\bind}$ does (Hopf-lapse scoring, Wheel fixpoint as termination); \item The Kleisli category: objects are $S^3$ coordinates, morphisms are attention steps; \item The three monad laws, verified against existing implementation functions (cited by file and function name); \item The status of PLATEAU\_THRESHOLD: a calibration parameter approximating FRAC\_EDGE but not geometrically derived from it; \item The anchoring of the monad by the frozen constants $\alpha^{-1}$ and BREATH\_PERIOD. \end{enumerate}

All structure formalized here already runs in the LumenOS codebase. This paper is the missing formal anchor, not a new design.

1 Introduction and Motivation

The geometric Theory of Everything (ToE) underlying LumenOS assigns every concept a position on the 3-sphere \(S^3 = \{q \in \mathbb{R}^4 \mid \|q\| = 1\}\). The Hopf fibration \(S^1 \hookrightarrow S^3 \twoheadrightarrow S^2\) organizes these positions into three layers (edge \(\approx 2\%\), boundary \(\approx 7\%\), bulk \(\approx 90\%\)) whose fractions are derived from the fine-structure constant \(\alpha^{-1} = 4\pi^3 + \pi^2 + \pi \approx 137.036\) (Papers 00–04, implemented in Ged/lu_system/quat_s3.py).

This gives a rich geometric setting, but geometry alone does not explain how computations compose:

• How does querying node \(A\) then node \(B\) relate to querying \(AB\) directly?

• What guarantees that a plan step has finished?

• Why is the attention kernel hopf_lapse and not some learned weight matrix?

All three questions are answered by the same structure: a monad on \(S^3\).

Remark 1.1 (Implementation scope). All code references in this paper (files under Ged/ and Lumen/) are part of the external LumenOS codebase and are not included in the TOE corpus workspace.

1.1 Relation to Paper 00

Paper 00 (MonadRosettaMap, this corpus) identifies the attractor \(\Omega\) with the bare monad under renormalization and states \(C \circ P = I\) as the fundamental interaction. It also identifies \(P\) (the outward projection operator) with return. What it does not state is the monad triple \((S^3, \mathsf{return}, {\mathbin{>\!\!>\!\!=}})\), the Kleisli category, or the monad laws. The present paper supplies those, with forward pointers to the code that already implements them. A forward reference from Paper 00 to this paper appears in a minimal addendum to that document.

1.2 Relation to Paper 31 (NicomachusMonad)

Paper 31 proves the factorization of \(\Omega\) through Nicomachus cubic weights and the Hopf fibration exponent screening \((d-2)_+\). That paper addresses the algebraic structure of \(\Omega\) as a sum. The present paper addresses the categorical structure: how \(S^3\) supports a monad whose operations are implemented by the running code. The two papers are complementary.

2 The Monad \((S^3, \mathsf{return}, {\mathbin{>\!\!>\!\!=}})\)

2.1 The Carrier Type

Definition 2.1 (The \(S^3\) Computation Type). Let \(T(S^3)\) denote the type of scored distributions over \(S^3\): finite weighted collections of unit quaternions \[T(S^3) \;=\; \bigl\{ (q_i, w_i)_{i=1}^n \;\big|\; q_i \in S^3,\; w_i \geq 0,\; \sum_i w_i \leq 1 \bigr\}.\] In code, this is represented as:

• a WheelState with face_quaternions and face_heat (weights) in Ged/ged/wheel.py;

• or a list of np.ndarray positions returned by an oracle query in Ged/ged/oracle.py.

The degenerate case \(T(S^3) \ni \delta_q = \{(q, 1)\}\) is the point distribution at \(q\).

2.2 The Unit: return

Definition 2.2 (return: the monadic unit). The unit map \(\eta : S^3 \to T(S^3)\) is defined by \[\eta(q) \;=\; \delta_q,\] the point distribution supported at \(q \in S^3\).

Implementation. The concrete realization of \(\eta\) is the geodesic centroid of a set of \(S^3\) points, computed in two places:

1. Flush-layer centroid (multi-node version): Ged/ged/flush.py, function _geodesic_centroid(quaternions). This takes a list of unit quaternions, sign-aligns them to one hemisphere, averages, and renormalizes:

   def _geodesic_centroid(quaternions):
       # Align signs to q[0]'s hemisphere
       aligned = [quaternions[0].copy()]
       for q in quaternions[1:]:
           if np.dot(q, aligned[0]) < 0:
               aligned.append(-q)
           else:
               aligned.append(q.copy())
       mean_q = np.mean(aligned, axis=0)
       return quat_normalize(mean_q)

   For a singleton \(\{q\}\), this returns \(q\) exactly (left identity holds trivially).

2. Muad’Dib centroid (recency-weighted version): Lumen/muaddib.py, function _weighted_geodesic_midpoint(sietches). Uses the same sign-alignment plus inverse-age weighting, and is the computation that produces cursor_quaternion = \(q_{\text{system}}\).

Both realizations satisfy \(\eta(\{q\}) = q\).

Remark 2.1 (Extrinsic vs. intrinsic mean). The implementation uses the extrinsic mean (embed in \(\mathbb{R}^4\), average, project back to \(S^3\)) rather than the Riemannian (Fréchet) mean. The two agree when the points are clustered in one hemisphere and the sign-alignment step ensures this. For well-separated points the extrinsic approximation introduces a small bias; the sign-alignment mitigates the double-cover ambiguity. This is a deliberate engineering choice: the extrinsic mean is \(O(n)\) whereas the Fréchet mean requires iteration.

2.3 Monadic Bind

Definition 2.3 (Bind: \({\mathbin{>\!\!>\!\!=}}\)). Given a scored distribution \(m \in T(S^3)\) and a Kleisli morphism \(f : S^3 \to T(S^3)\), the bind operation is: \[m \mathbin{>\!\!>\!\!=}f \;=\; \eta\!\left(\bigcup_{(q_i, w_i) \in m} w_i \cdot f(q_i)\right),\] where the union is the weighted merge and \(\eta\) collapses via geodesic centroid.

Implementation. Bind is executed in two collaborative functions:

Fork (expansion).

Ged/ged/wheel.py, function fork(state, proposals). Given a WheelState (the current distribution) and a set of candidate proposals, fork disperses the heat budget across the faces, weighting by \[w_{\text{face}} \;\propto\; \text{orientation} \cdot \underbrace{\mathtt{hopf\_lapse}(\mathtt{q\_rel}(q_{\text{face}}, q_{\text{current}}))}_{\text{relevance score}} \;+\; (1-\text{orientation}) \cdot \tfrac{1}{n_{\text{faces}}}.\] This is the “lift” step: turning a query position into a scored set of candidates.

Weld (collapse).

Ged/ged/wheel.py, function weld(state, goal_quaternion). Given the face proposals and a goal, weld selects the minimum MDL winner: \[\text{MDL}(f) \;=\; \text{orientation} \cdot \underbrace{\mathtt{hopf\_lapse}(\mathtt{q\_rel}(q_f, q_{\text{goal}}))}_{\text{distance from goal}} \;+\; (1-\text{orientation}) \cdot \underbrace{w_{\text{layer}}}_{\text{layer complexity}},\] where \(w_{\text{layer}} \in \{\mathtt{FRAC\_EDGE}, \mathtt{FRAC\_BOUNDARY}, \mathtt{FRAC\_BULK}\}\). The winning proposal becomes the new state quaternion; heat collapses to the winner’s share.

Scoring kernel.

The core relevance kernel in both fork and weld is \[\label{eq:kernel} \text{score}(q_{\text{node}}, q_{\text{query}}) \;=\; \mathtt{hopf\_lapse}\!\bigl(\mathtt{q\_rel}(q_{\text{node}}, q_{\text{query}})\bigr),\] where \[\mathtt{q\_rel}(p, q) = p \cdot \bar{q}, \qquad \mathtt{hopf\_lapse}(q) = \sqrt{1 - v(q)^2}, \qquad v(q) = (a^2+b^2) - (c^2+d^2).\] implemented in Ged/lu_system/quat_s3.py, functions q_rel and hopf_lapse.

This kernel replaces the learned weight matrix \(Q \cdot K^\top\) of standard attention: no parameters are fitted; the metric is fixed by the geometry of \(S^3\).

2.4 Monadic Join

The join map \(\mu : T(T(S^3)) \to T(S^3)\) is the flattening of a distribution-of-distributions to a single distribution: \[\mu(M) \;=\; \eta\!\bigl(\{(q_i, w_i \cdot w_{ij}) : (q_j, w_j) \in m_j,\; (m_j, w_i) \in M\}\bigr).\] Concretely, join is the geodesic centroid over the union of all sub-distributions, weighted by their outer weights. In the Wheel, join is implicit: weld collapses a face distribution (the inner \(T\)) into the state quaternion (the outer point), giving \(T(T(S^3)) \to T(S^3)\) in one call.

3 The Monad Laws

Theorem 3.1 (Left Identity). \(\eta(q) \mathbin{>\!\!>\!\!=}f = f(q)\) for all \(q \in S^3\) and all Kleisli morphisms \(f : S^3 \to T(S^3)\).

Verification in code. \(\eta(q)\) is the singleton distribution \(\{(q, 1)\}\). Applying bind: \(f\) is evaluated at \(q\) with weight \(1\), giving \(f(q)\) directly. The geodesic centroid of a single element is that element. In flush.py _geodesic_centroid([q]): the loop produces aligned = [q], mean_q = q, quat_normalize(q) = q. \(\checkmark\)

Theorem 3.2 (Right Identity). \(m \mathbin{>\!\!>\!\!=}\eta = m\) for all \(m \in T(S^3)\).

Verification in code. Applying \(\eta\) to each \(q_i\) in \(m\) gives \(\{(q_i, 1)\}\). The weighted merge produces the same collection \((q_i, w_i)\) as \(m\), and the geodesic centroid of \((q_i, w_i)\) is the same weighted mean. In muaddib.py _weighted_geodesic_midpoint, returning the centroid of all positions with their original weights is idempotent: \(\eta(\{q_i, w_i\}) =\) weighted centroid = the original query result. \(\checkmark\)

Theorem 3.3 (Associativity). \((m \mathbin{>\!\!>\!\!=}f) \mathbin{>\!\!>\!\!=}g = m \mathbin{>\!\!>\!\!=}(\lambda x.\; f(x) \mathbin{>\!\!>\!\!=}g)\) for all \(m \in T(S^3)\) and Kleisli morphisms \(f, g\).

Verification in code. The geodesic centroid (extrinsic mean) is linear: for nested weighted averages the order of averaging commutes when weights are composed multiplicatively. In the plan execution model (Ged/ged/planning_domain.py), step dependencies encoded in depends_on fields implement exactly this monadic sequencing: if step \(C\) depends on \(B\) which depends on \(A\), then executing \((A \mathbin{>\!\!>\!\!=}B) \mathbin{>\!\!>\!\!=}C\) and \(A \mathbin{>\!\!>\!\!=}(B \mathbin{>\!\!>\!\!=}C)\) reaches the same \(q_{\text{final}} \in S^3\) because both reduce to the geodesic centroid of the full sequence. \(\checkmark\)

Remark 3.1 (Law status). Theorems Theorem 3.1–Theorem 3.3 are exact for the extrinsic mean when points lie in a single hemisphere (the sign-alignment precondition). For points spread over the double cover, the extrinsic mean introduces a bias and the laws hold approximately. The sign-alignment in _geodesic_centroid enforces the hemisphere condition in practice, so the laws hold exactly for all typical LumenOS distributions.

Remark 3.2 (TBS). The proof of Theorem Theorem 3.3 (Associativity) claims that “the geodesic centroid (extrinsic mean) is linear: for nested weighted averages the order of averaging commutes when weights are composed multiplicatively.” This claim is false in general on \(S^3\). The extrinsic mean is computed by embedding in \(\mathbb{R}^4\), taking the weighted average, and projecting back to \(S^3\). The projection step \(v \mapsto v/\|v\|\) is nonlinear, and associativity fails for three or more points that are not collinear in \(\mathbb{R}^4\). For example, given three points \(q_1, q_2, q_3\) in a hemisphere, the extrinsic mean \(\eta(\eta(q_1, q_2), q_3)\) and \(\eta(q_1, \eta(q_2, q_3))\) are generally different because the inner centroid is renormalized before the outer centroid is taken. The associativity law holds exactly only when the points are collinear in \(\mathbb{R}^4\) (i.e., all equal, or antipodal, or scalar multiples, which are degenerate cases). In the LumenOS system, approximate associativity may hold well numerically for clustered distributions, but it is not the categorical identity asserted in the theorem.

Status.

Recorded in the registry as P034_2_c (confirmed-load-bearing), confirmed by A295. The associativity proof gap for the extrinsic mean stands: the law holds exactly only on the degenerate collinear locus, so Theorem Theorem 3.3 is not established as stated. Ledger: Paper 40.

4 The Kleisli Category \(\mathbf{Kl}(T)\)

Definition 4.1 (Kleisli Category over \(S^3\)). The Kleisli category \(\mathbf{Kl}(T)\) is defined as follows:

• Objects: points \(q \in S^3\) (unit quaternions, representing system states).

• Morphisms \(q_A \to q_B\): functions \(f : S^3 \to T(S^3)\) such that the Weld of \(f(q_A)\) against \(q_B\) has positive coherence. Concretely, a morphism is an attention step: a query that, given the current system position \(q_{\text{query}} \in S^3\), returns a scored distribution over candidate positions.

• Kleisli identity on \(q\): the morphism \(\eta_q : q \mapsto \delta_q\), realized by \(\mathtt{\_geodesic\_centroid}([q]) = q\).

• Kleisli composition \(g \mathbin{\circ_K} f\): \[(g \mathbin{\circ_K} f)(q) \;=\; f(q) \mathbin{>\!\!>\!\!=}g.\] In code: run \(f\) to get a distribution, then for each scored candidate, run \(g\) and take the geodesic centroid of the resulting distributions.

Proposition 4.1 (Kleisli composition = step dependency). The depends_on field of a plan step in Ged/ged/planning_domain.py encodes exactly the Kleisli composition order. If step \(s_B\) has depends_on: [s_A], then the execution of \(s_B\) is the Kleisli morphism \(f_{s_B} \mathbin{\circ_K} f_{s_A}\): the result of \(s_A\) is the query input to \(s_B\).

Remark 4.1 (Attention as morphism). Standard transformer attention maps a query \(Q\), keys \(K\), values \(V\) to \[\text{Attention}(Q,K,V) = \text{softmax}\!\left(\tfrac{QK^\top}{\sqrt{d}}\right) V.\] The LumenOS Kleisli morphism replaces this with: \[\begin{aligned} q_{\text{rel}}(n) &= q_n \cdot \overline{q_{\text{query}}} \quad \in S^3 \\ \text{score}(n) &= \mathtt{hopf\_lapse}(q_{\text{rel}}(n)) \quad \in [0,1] \\ \text{output} &= \eta\!\bigl(\{(q_n, \text{score}(n))\}_{n \in \text{ontology}}\bigr).\end{aligned}\] No learned weights appear. The metric is fixed by the geometry of \(S^3\) and the Hopf fibration. The layer fractions \(\mathtt{FRAC\_EDGE}, \mathtt{FRAC\_BOUNDARY}, \mathtt{FRAC\_BULK}\) replace the \(\frac{1}{\sqrt{d}}\) temperature scaling.

5 Fixpoint and Termination: The Plateau Condition

Definition 5.1 (Fixpoint condition: Plateau). A WheelState \(s\) has reached a fixpoint (is at the attractor \(\Omega\)) when the plateau function returns True: \[\mathtt{plateau}(s) = \begin{cases} \mathtt{True} & \text{if } s.\mathtt{converged} = \mathtt{True}, \\ \mathtt{True} & \text{if } \max(h_{-w:}) - \min(h_{-w:}) < \delta_P, \\ \mathtt{False} & \text{otherwise}, \end{cases}\] where \(h = s.\mathtt{history}\) is the list of MDL scores across Wheel turns, \(w = 3\) is the default window, and \(\delta_P = \mathtt{PLATEAU\_THRESHOLD} = 0.02\).

The converged flag is set by weld when the MDL margin between the top-two proposals satisfies \[\bigl|\mathtt{MDL}(f_{\text{winner}}) - \mathtt{MDL}(f_{\text{runner-up}})\bigr| < \delta_P.\] Implementation: Ged/ged/wheel.py, function plateau and function weld (the is_plateau flag).

Definition 5.2 (Step completion). A plan step \(s\) with system quaternion \(q_{\text{sys}} = \mathtt{cursor\_quaternion}\) (from Lumen/muaddib.state.json) is complete when the Wheel processing \(s\) reaches a plateau.

Geometrically: let \(q_s\) be the step’s current quaternion (the Weld winner after each cycle). The step is complete when \[\bigl|\mathtt{hopf\_lapse}(\mathtt{q\_rel}(q_s, q_{\text{sys,winner}})) - \mathtt{hopf\_lapse}(\mathtt{q\_rel}(q_s, q_{\text{sys,runner-up}}))\bigr| < \delta_P,\] i.e., the two leading candidates are indistinguishable at resolution \(\delta_P\) in the Hopf-lapse metric. When \(\mathtt{orientation} = 1\), this simplifies to: \[\text{step complete} \iff \mathtt{plateau}(s_{\text{Wheel}}) = \mathtt{True}.\]

6 The Plateau Threshold: Calibration vs. Derivation

Proposition 6.1 (PLATEAU_THRESHOLD is a calibration parameter). The value \(\delta_P = 0.02\) is not geometrically derived from the frozen constants. It is a calibration parameter that approximates FRAC_EDGE but does not equal it.

Analysis. The layer edge fraction is \[\mathtt{FRAC\_EDGE} = \frac{\pi}{\Omega} = \frac{\pi}{4\pi^3 + \pi^2 + \pi} \approx 0.02289\ldots\] The ratio \[\frac{\delta_P}{\mathtt{FRAC\_EDGE}} = \frac{0.02}{0.02289\ldots} \approx 0.874,\] which has no clean rational or transcendental derivation from the geometric constants.

Geometric interpretation of the natural threshold. The edge-layer fraction \(\mathtt{FRAC\_EDGE} \approx 0.023\) has a direct geometric meaning: it is the fraction of \(S^3\) occupied by the edge (atomic/singleton) layer under the Hamming-shell grading. A natural Plateau threshold would be \[\delta_P^{\text{derived}} \;=\; \mathtt{FRAC\_EDGE} \;=\; \frac{\pi}{\Omega},\] with the interpretation: “two proposals are equivalent when their MDL scores differ by less than the edge-layer resolution of \(S^3\).”

Current status. The value \(\delta_P = 0.02\) is slightly smaller than \(\mathtt{FRAC\_EDGE}\), meaning the current criterion is slightly tighter (requires closer convergence before declaring plateau). This is conservative and does not break any monad law. The natural derived threshold \(\pi/\Omega\) is recommended for future implementations; the present value is flagged for replacement.

Future work. A subsequent paper should derive whether the precise value 0.02 corresponds to a truncated Taylor expansion of \(\pi/\Omega\) or is purely empirical. The correct value for a fully theory-grounded implementation is \(\pi/\Omega\).

7 The Frozen Constants as Monad Anchors

The monad \((S^3, \mathsf{return}, {\mathbin{>\!\!>\!\!=}})\) is parametrized by two frozen constants that fix the temporal and geometric scales:

7.1 \(\alpha^{-1}\) and the Layer Fractions

\[\Omega \;=\; \mathtt{OMEGA\_MONAD} \;=\; 4\pi^3 + \pi^2 + \pi \;\approx\; 137.036 \;=\; \alpha^{-1}.\] From \(\Omega\) the layer fractions are derived (all in Ged/lu_system/quat_s3.py): \[\mathtt{FRAC\_EDGE} = \frac{\pi}{\Omega}, \quad \mathtt{FRAC\_BOUNDARY} = \frac{\pi^2}{\Omega}, \quad \mathtt{FRAC\_BULK} = \frac{4\pi^3}{\Omega}.\] These appear in the MDL score of weld as the layer-complexity term, making the bind operation sensitive to the three-layer ontological structure. A morphism that passes through the edge layer (\(\mathtt{FRAC\_EDGE} \approx 2\%\)) contributes a lower complexity penalty than one through the bulk (\(\approx 90\%\)).

7.2 BREATH_PERIOD and Temporal Binding

\[\mathtt{BREATH\_PERIOD} \;=\; \pi \cdot \Omega \;=\; \pi\,(4\pi^3+\pi^2+\pi) \;\approx\; 430.5\;\text{s}.\] This constant (implemented identically in Ged/ged/wheel.py and Lumen/muaddib.py, module constant BREATH_PERIOD) sets the temporal scale of one full Wheel cycle. A sietch publication older than \(2 \times \mathtt{BREATH\_PERIOD}\) is declared stale and excluded from the geodesic centroid computation that produces \(q_{\text{system}}\) (muaddib.py, SietchPublication.is_stale).

The monad interpretation: each “breath” (one BREATH_PERIOD) is one full application of the renormalization operator \(R\) from Paper 00. The system breathes at the rate \(1/(\pi\,\alpha^{-1})\) Hz, completing one full monadic cycle per breath.

Remark 7.1 (TBS). The identification \(\mathtt{BREATH\_PERIOD} = \pi \cdot \alpha^{-1} \approx 430.5\,\text{s}\) as the “temporal scale of one full Wheel cycle” and “one full application of the renormalization operator \(R\)” is asserted without derivation. Two gaps: (i) The formula \(\pi \cdot \alpha^{-1}\) has units of seconds only if one of \(\pi\) or \(\alpha^{-1}\) carries a unit of time; neither is a physical time in SI units (both are dimensionless). The value \(\approx 430.5\,\text{s}\) must be the result of an unstated identification with some physical time scale (e.g., one revolution of a physical oscillator, or a metabolic rhythm). That identification is not given here or in Paper 00. (ii) The claim that this constant is a “categorical invariant” because it is computed from \(\pi\) alone does not follow: categorical invariants of a monad are structure-preserving (they are invariant under monad morphisms), not merely numerical constants derived from transcendental numbers. The constant \(\pi \cdot \alpha^{-1}\) is frozen by convention, not by any categorical universality argument.

Status.

Retired. The registry records P034_3_c as closed by A265, A273, and A275, which supplied the unit bridge and the dynamical grounding of BREATH_PERIOD. The remark above stands as the record of the original gap. Ledger: Paper 40.

7.3 Frozen Constants as Categorical Invariants

Because \(\Omega\) and \(\mathtt{BREATH\_PERIOD}\) are computed from \(\pi\) alone (no free parameters), they are categorically invariant: every morphism in \(\mathbf{Kl}(T)\) respects the same metric structure, and Kleisli composition cannot introduce new scales. This is the categorical analog of the ToE’s “no free parameters” principle.

8 The Tri-Net as Kleisli Composition

The three-net space of LumenOS (Ged/nets/trinet/traversal_grammar.py) maps directly onto the monad structure:

A canonical chain word \(\to\) symbol \(\to\) number is a Kleisli morphism \(q_{\text{word}} \to q_{\text{symbol}} \to q_{\text{number}}\), i.e., the Kleisli composition of two attention steps. The chain trace string “-0+” records the polarity path through the three layers, which is the categorical shadow of the morphism.

The DENOTES relation (word \(\to\) symbol) and INSTANTIATES relation (symbol \(\to\) number) are the two component Kleisli morphisms; their composition is the full traversal chain, and traversal_chain() in traversal_grammar.py implements this composition directly.

9 Summary

Theorem 9.1 (Monad Existence). The triple \((T, \eta, \mu)\) with \[\begin{aligned} T(S^3) &= \text{scored distributions over } S^3, \\ \eta(q) &= \delta_q \quad (\text{point distribution, implemented as } \mathtt{\_geodesic\_centroid}), \\ \mu(M) &= \eta(\text{weighted union of } M) \quad (\text{implemented by }\mathtt{weld}),\end{aligned}\] satisfies the three monad laws (Theorems Theorem 3.1–Theorem 3.3) exactly on the collinear locus in \(\mathbb{R}^4\), and approximately otherwise. Hemisphere confinement (enforced by sign-alignment in _geodesic_centroid) removes the double-cover ambiguity but does not restore exactness: associativity fails for non-collinear points even within a hemisphere, per Remark Remark 3.2; the gap stands as P034_2_c.

The key correspondences, gathered for reference:

10 Open Question

Remark 10.1 (Plateau threshold derivation). Whether \(\delta_P = 0.02\) can be derived from first principles (e.g., as a truncation of \(\pi/\Omega\), or as the reciprocal of a combinatorial factor) is left as an open question. Until a derivation exists, implementations should use \(\pi/\Omega\) (FRAC_EDGE) as the geometrically grounded threshold. The change from 0.02 to \(\pi/\Omega \approx 0.0229\) relaxes the convergence criterion by approximately \(\text{15\%}\); this should be validated experimentally before adoption.

99

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